We now suppose \({\operatorname{an}}_{\mathfrak {m}}\)  corresponds to a reducible residual representation, so that \(\mathfrak {m}\)  is the Eisenstein ideal generated by 2 and \(T_{\ell }\)  for \(\ell \nmid N\)  (including \(\ell =2\) ). We claim that \(T_2\)  is already in \({\operatorname{an}}_{\mathfrak {m}}\) . This is because by [1]}, the Eisenstein ideal of the full Hecke algebra is generated by \(1+\ell -T_{\ell }\)  for any good prime. So by completeness, \(T_2-3\)  and therefore \(T_2\)  can be written as a power series in \(T_\ell -\ell -1\) .
